Global order from local sources

Abstract

This article contains introductions to three open problems of significant research interest, taken from number theory, logic, and condensed matter physics. All three problems will be shown to have at their core special cases of one simply-stated optimization problem. Our goal is to use the intuition gained from these three perspectives to direct attention to this common core, which constitutes, in fact, one problem of remarkable depth and importance. We will also show that some of the tools developed in the separate problems are of real value in the others.